## ag.algebraic geometry – Evaluate the following algebraic expressions. Show all of your steps

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## ag.algebraic geometry – Universal bundles over algebraic stacks

$$DeclareMathOperatorel{ell}$$In the topological case, given a group $$G$$, we can define the classifying space $$BG$$ for principal $$G$$-bundles and there is a universal $$G$$-principal bundle $$EG to BG$$ classified by the identity morphism $$BG to BG$$ such that any other principal bundle over $$X$$ is given by a pullback of the universal bundle along some map $$X to BG$$. Is there an analog construction for algebraic stacks?

That is, take some algebraic stack classifying a family of objects over a given scheme, e.g. the moduli stack of elliptic curves. Is there any way to make sense of “the universal elliptic curve classified by the identity morphism $$mathcal{M}_{el} to mathcal{M}_{el}$$“? My problem is I’m not sure how to evaluate an algebraic stack on an algebraic stack – we can glue the value of a stack along a cover of a scheme, but is there any reason to expect a similar property for covers of algebraic stacks? Will the resulted object be an algebraic stack?

I’ve found this “universal bundle” mentioned in this nLab article, where they even take its bundle of differential forms, but I couldn’t find any reference or explanation, so any reference would be very welcome.

## ag.algebraic geometry – A modern proof of the Verlinde formula?

Let $$G$$ be a semisimple algebraic group and $$Sigma$$ a smooth proper curve. Then $$text{Bun}_G(Sigma)$$ comes equipped with a line bundle $$mathcal{L}$$ which generates the torsion free part of $$text{Pic}text{Bun}_G(Sigma)$$ (e.g. in type A or C it’s the determinant bundle or in type B or D the Pfaffian bundle). Then the Verlinde formula is an explicit formula for
$$H^0(text{Bun}_G(Sigma),mathcal{L}^{otimes k}),$$
intimately related to fusion products.

There are lots of nice proofs of the Verlinde formula published in the 1990s, e.g. Beauville’s Conformal blocks, fusion rules and the Verlinde formula. However, they are all quite algebraic which makes it hard (at least for me) to understand what’s going on. Given how much better the geometric side has been understood in recent decades (e.g. fusion and the BD Grassmannian), is there written up anywhere a slightly cleaner/more geometric proof of the Verlinde formula?

## ag.algebraic geometry – Open immersion of affinoid adic spaces

If $$R$$ and $$S$$ are complete Huber rings with $$varphi: R to S$$ a continuous map, then is it true in general that if $$mathrm{Spa}(S, S^circ) to mathrm{Spa}(R, R^circ)$$ is an open immersion of adic spaces (here $$S^circ$$ and $$R^circ$$ are the power-bounded subrings) then $$mathrm{Spec}(S) to mathrm{Spec}(R)$$ is injective?

For example, this is true if $$R$$ and $$S$$ both have the discrete topology, because if $$frak p$$ and $$frak q$$ are two prime ideals in $$S$$ which are equal after restricting to $$R$$ then $$(frak p, |cdot|_{rm triv})$$ and $$(frak q, |cdot|_{rm triv})$$ (trivial valuations), which are both points in $$mathrm{Spa}(S,S)$$, restrict to the trivial valuation on $$R/varphi^{-1}(frak p)$$.

But I’m not sure how generally to expect that this is true.

## ag.algebraic geometry – Singular hypersurface section

Suppose $$X$$ is a smooth projective surface over complex numbers in $$mathbb{P}^n$$. Let $$text{Spec}(R)$$ be an affine open set of $$X$$. Then $$R$$ is a finitely generated $$mathbb{C}$$ algebra of dimension $$2$$. Let $$Z:= {p_1, …p_k}$$ be a set of distinct closed points in $$text{Spec}(R)$$. Since $$X$$ is smooth, there are regular parameters $$x_i, y_i$$ such that the maximal ideal $$m_{p_i}$$ is generated by $$x_i$$ and $$y_i$$. Let $$I$$ be ideal defined by $$m^{‘}_{p_1}m^{‘}_{p_2}…m^{‘}_{p_k}$$, where $$m^{‘}_{p_i} = langle x_i^2, y_irangle$$. Let $$Z^{‘}$$ be the subscheme defined by $$I$$.

What is the geometric interpretation of $$Z^{‘}$$ in terms of $$Z$$?

Is $$Z^{‘}$$ locally complete intersection?

Suppose $$Y in H^0(I_{Z^{‘}}(d))$$ is a degree $$d$$ hypersurface section. Then can we say that $$Y$$ is singular along $$Z$$ ?

## ag.algebraic geometry – The period map and the Kodaira–Spencer map

Let $$f : X to B$$ be a non-isotrivial holomorphic submersion with connected fibres between compact Kähler manifolds of positive relative dimension. Suppose the fibres of $$f$$ are Calabi–Yau $$(c_{1,mathbb{R}}=0$$) or canonically polarised ($$c_1<0$$). The differential of the moduli map $$mu : B^{circ} to mathcal{M}$$ is the Kodaira–Spencer map $$tau = dmu$$, measuring in the complex structure of the smooth fibres of the family. The period map $$p : B^{circ} to D$$ is a holomorphic map which measures the variation of the Hodge decomposition of the fibres.

Is there a relationship between the Kodaira–Spencer map and the period map?

## ag.algebraic geometry – Looking for the exact and the precise statement of Ogus conjecture

I have been looking for several weeks for the exact and the precise statement of Ogus conjecture, but, I cannot find it.
The only book which made me discover the statement of this conjecture is that of Yves André: Théorie des motifs. See here, http://tomlr.free.fr/Math%E9matiques/Andre,%20Y%20-%20Une%20Introduction%20aux%20Motifs%20%28SMF%202004%29.pdf, pages, $$79$$ and $$80$$.

To understand how this conjecture is formulated, the author of this book directs us to a paper of Ogus, the holder of this conjecture, which is entitled, Hodge Cycles and Crystalline Cohomology.

The paper can be found here, https://www.jmilne.org/math/Books/DMOS.pdf, page, $$359$$, in the introduction.

The statement of Ogus conjecture is not very clear. I formulated it as follows, following my efforts to understand its statement.

Here is the statement that I propose,

Let $$k$$ be a number field.

Let $$R$$ be an étale $$mathbb {Z}$$ -algebra.

Let $$X$$ be a smooth projective $$R$$ -scheme.

Let $$R’ supseteq R$$ be another étale $$mathbb{Z}$$ – algebra.

Let $$s$$ be a closed point of $$mathrm{Spec} R ‘$$, and let $$W$$ be the completion of $$R’$$ in $$s$$.

We have an isomorphism, $$H_ {mathrm{dR}}^{i} (X / R) otimes_k W simeq H_{mathrm{cris} }^{i} (X (s) / W )$$

$$H_{mathrm{cris}}^{i} (X(s) / W)$$ is a $$F_ {displaystyle v}$$ – crystal, therefore, equipped with the Frobenius $$F_{ displaystyle v} : H_{mathrm{cris}}^{i} (X(s) / W) to H_{mathrm{cris}}^{i} (X(s) / W)$$ defined by, $$F_{displaystyle v} (z) = p^r z$$.

So, we can pass this Frobenius $$F_{displaystyle v}$$, to $$H_{mathrm{dR}}^{i} (X / R) otimes_k W$$ by this isomorphism.

Let the integral class cycle map (i.e., on $$mathbb {Z}$$), be defined by,
$$mathrm{cl}_X : mathcal{Z}_{sim}^{i} (X) to displaystyle bigcup_{v in I} displaystyle Big (H_{mathrm{dR}}^{i} (X / R) otimes_k W displaystyle Big)^{textstyle F_{v}},$$ where, $$Big (H_{mathrm{dR}}^{i} (X / R) otimes_k W Big)^{textstyle F_{v}}$$ is the $$F_{displaystyle v}$$ – crystal of $$F_{textstyle v }$$ – invariants.

$$I$$ is the collection of the closed points $$s$$ of $$mathrm{Spec} R’$$.

So, Ogus conjecture asserts that, the rational class cycle map (i.e., over $$mathbb{Q}$$), which is $$mathrm{cl}_X otimes mathbb{Q}$$, as follow, $$mathrm{cl}_X otimes mathbb{Q} : mathcal{Z}_{sim}^{i} (X) otimes_{ mathbb{Z} } mathbb{Q} to displaystyle bigcup_{v in I} displaystyle Big (H_{mathrm{dR}}^{i} (X / R) otimes_k W displaystyle Big)^{textstyle F_{v}} otimes_{ mathbb{Z} } mathbb{Q}$$

is surjective?

So, is that right ?

Can you correct that statement for me to see if I got it right?

How is $$W$$ defined ?

Does $$W$$ vary when the closed point $$s$$ of $$mathrm{Spec} R’$$ varies ?

See here, Berthelot-Ogus comparison isomorphism for others interesting informations.

## ag.algebraic geometry – How do I determine a 4th location that is equidistant to three other locations

ag.algebraic geometry – How do I determine a 4th location that is equidistant to three other locations – MathOverflow

## ag.algebraic geometry – Negative curves on surfaces

Let $$f:Xrightarrowmathbb{P}^1$$ be a family of surfaces over $$mathbb{C}$$. Assume that for $$yinmathbb{P}^1$$ general $$X_y = f^{-1}(y)$$ is a smooth surface. Let $$X_{eta}$$ be the generic fiber of $$f$$ and $$overline{X}_{eta} := X_{eta}times_{Spec(mathbb{C}(t))}Spec(overline{mathbb{C}(t)})$$, where $$overline{mathbb{C}(t)}$$ is the algebraic closure of $$mathbb{C}(t)$$.

Assume that in $$overline{X}_{eta}$$ there is a curve of negative self-intersection $$-a$$. Does there is a curve of negative self-intersection $$-a$$ on the fiber $$X_y = f^{-1}(y)$$ for $$yinmathbb{P}^1$$ general?

## ag.algebraic geometry – Berthelot-Ogus comparison isomorphism

On the following link, https://perso.univ-rennes1.fr/matthieu.romagny/exposes/Theoremes_de_comparaison_et_representatons_galoisiennes.pdf, page, $$2$$, the Berthlot-Ogus isomorphism theorem is stated as follows,

We have a canonical isomorphism, $$rho_{mathrm{cris}} : H_{mathrm{cris}}^{i} (X) otimes_{K_ {0}} K to H_{mathrm{dR}}^{i} (mathscr{X}) otimes_{O_K} K = H_{mathrm{dR}}^{i} (X) qquad (1)$$
On the other hand, on the following pdf, https://www.jmilne.org/math/Books/DMOS.pdf, page, $$359$$, the author claims that there is an isomorphism, $$H_ {mathrm{DH}} (X) otimes W simeq H_{mathrm{cris}} (X (s) / W) qquad (2)$$.

What is the difference between the two isomorphisms $$(1)$$ and $$(2)$$ ?

Are the two isomorphisms the same ?

Thanks in advance for you help.